I have recently tried to understand how a Tesla coil works. It is very similar to a transformer which is fundamentally two inductors interlinked by an iron core and serves to increase or decrease an input AC voltage (vice versa for the AC current).

Tesla coils, which can wirelessly turn on nearly fluorescent lights, are said to be resonant transformers. In what sense are ordinary capacitors not resonant?

Tesla coil uses air core.
So basically it is not that much different from normal transformer.

Original Tesla coil uses spark gap to induce signal in primarily coil.
Resonant frequency is achieved with LC curcuit which is conected to the ground on one end and regulation of distance on spark gap contacts
This was only way to achieve high frequencies back than.
Now days it is fairly easier and more precise to use investor circuit for inducing primary coil.

As someone who owns a tesla coil and loves making sparks, this was the best layman explanation of how a tesla coil works. I have personally had trouble explaining to non technical friends and family how tesla coils work, since I usually start explaning how lc circuits work, then realize I have to explain voltage first. Good job!